Question

Determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations. If so, find the equations. $$ u_{x x}+u_{y y}+x u=0 $$

Short answer

Yes, we can use the method of separation of variables on the given PDE. After assuming a product form of the solution, differentiating with respect to x and y, and substituting these into the PDE, we separated the equation into terms with functions of x and y alone. Finally, we obtained a pair of ODEs: $$ -X'' - xX = \lambda X \quad \text{and} \quad Y'' =\lambda Y $$ These ODEs can be used to find a solution to the given PDE using the separation of variables method.

Step-by-step solution

Assume a product form of the solution

We need to assume a product form of the solution as given below: $$ u(x, y) = X(x)Y(y) $$

Compute the second derivatives of u with respect to x and y

Using the product form of the solution, we will compute the second derivatives of u with respect to x and y: $$ u_{xx} = \frac{\partial^2 u}{\partial x^2} = \frac{\partial^2}{\partial x^2}(XY) = X''Y \quad (\text{where } X'' = \frac{d^2X}{dx^2}) $$ and $$ u_{yy} = \frac{\partial^2 u}{\partial y^2} = \frac{\partial^2}{\partial y^2}(XY) = XY'' \quad (\text{where } Y'' = \frac{d^2Y}{dy^2}) $$

Substitute the second derivatives and product form of the solution into the PDE

Now, we will substitute the assumed product form of the solution and the computed second derivatives into the given PDE: $$ X''Y + XY'' + xXY = 0 $$

Check if we can separate the equation into terms with functions of x and y alone

We will now try to rewrite the equation as a sum of terms with functions of x alone and y alone: $$ X''Y + xXY = - XY'' $$ We may divide both sides of the equation by $$-XY$$: $$ \frac{-X''}{X} - x= \frac{Y''}{Y} $$ On the left-hand side, we have terms that depend on x alone, while on the right-hand side, we have terms that depend on y alone. These two sides must be equal to a constant, say $$\lambda$$, to satisfy the equation for all x and y: $$ \frac{-X''}{X} - x = \lambda = \frac{Y''}{Y} $$ We have successfully separated the equation into two ODEs: $$ -X'' - xX = \lambda X \quad \text{and} \quad Y'' =\lambda Y $$ These ODEs can be used to find a solution to the given PDE using the separation of variables method.

Key concepts

Partial Differential Equations

Partial Differential Equations (PDEs) are mathematical equations that involve functions of several variables and their partial derivatives. They are used to describe various phenomena in fields like physics, engineering, and finance.
For example, the heat equation and wave equation are classic PDEs. Generally, PDEs can be quite complex due to the involvement of more than one variable, and finding solutions often requires specialized methods.
In our problem, a given PDE is: \[u_{xx}+u_{yy}+xu=0\] This involves a function \(u(x, y)\) that depends on two variables—\(x\) and \(y\). The equation includes partial derivatives \(u_{xx}\) and \(u_{yy}\).
The main goal is to simplify such PDEs into something more manageable, like a pair of ordinary differential equations by using techniques like Separation of Variables.

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are mathematical equations involving functions of a single variable and their derivatives. Unlike PDEs, ODEs contain only ordinary derivatives. They are easier to solve since they involve just one independent variable.
In the context of the original problem, once we employ the method of Separation of Variables, the given PDE is transformed into a pair of ODEs: - \(-X'' - xX = \lambda X \) - \(Y'' = \lambda Y \)
Here, \(X(x)\) and \(Y(y)\) are functions of \(x\) and \(y\) separately, and \(\lambda\) is a constant parameter that arises from the separation process.
These ODEs can then be solved using various standard techniques specific to ordinary differential equations, which aim to find the individual functions \(X(x)\) and \(Y(y)\), comprising the solution to the partial differential equation.

Product Solution Form

The Product Solution Form is a useful approach when solving PDEs using the method of Separation of Variables. This method assumes that the solution can be expressed as a product of functions, each depending on only one of the independent variables.
In our problem, the solution is assumed to take the form \[u(x, y) = X(x)Y(y)\] This assumption allows us to express the partial derivatives of \(u\) in terms of the derivatives of \(X(x)\) and \(Y(y)\).
With the product form, the partial derivatives can be simplified as follows: - \(u_{xx} = X''Y\)- \(u_{yy} = XY''\)
By substituting these into the original PDE and equating the expressions by separating variables, we effectively break down the problem into two simpler ODEs. This product approach is a central part of the Separation of Variables technique, making it possible to reduce the complexity of PDEs.